Problem: Let $f$ be a 3D scalar field. Is the expression $\nabla \times (\nabla f)$ a scalar field, a vector field, or undefined? Choose 1 answer: Choose 1 answer: (Choice A) A Scalar field (Choice B) B Vector field (Choice C) C Undefined
Explanation: The gradient, which takes a scalar field and gives a vector field of the same dimension, can be written in two ways: $\text{grad}(f) = \nabla f$ The 3D curl, which takes a vector field and gives a vector field, can also be written in two ways: $\text{curl}(F) = \nabla \times F$ Therefore, $\nabla \times (\nabla f)$ is the curl of the gradient of a 3D scalar field. The gradient of a 3D scalar field is a 3D vector field. The curl of a 3D vector field is a vector field. The expression $\nabla \times (\nabla f)$ is a vector field.